In: Statistics and Probability
Many regions in North and South Carolina and Georgia have experienced rapid population growth over the last 10 years. It is expected that the growth will continue over the next 10 years. This has motivated many of the large grocery store chains to build new stores in the region. The Kelley’s Super Grocery Stores Inc. chain is no exception. The director of planning for Kelley’s Super Grocery Stores wants to study adding more stores in this region. He believes there are two main factors that indicate the amount families spend on groceries. The first is their income and the other is the number of people in the family. The director gathered the following sample information.
Family | Food | Income | Size | |||||
1 | $ | 5.26 | $ | 73.98 | 5 | |||
2 | 4.08 | 54.90 | 2 | |||||
3 | 5.76 | 59.12 | 4 | |||||
4 | 3.48 | 52.02 | 1 | |||||
5 | 4.20 | 65.70 | 2 | |||||
6 | 4.80 | 53.64 | 4 | |||||
7 | 4.32 | 79.74 | 3 | |||||
8 | 5.04 | 68.58 | 4 | |||||
9 | 6.12 | 165.60 | 5 | |||||
10 | 3.24 | 64.80 | 1 | |||||
11 | 4.80 | 138.42 | 3 | |||||
12 | 3.24 | 125.82 | 1 | |||||
13 | 6.45 | 77.58 | 7 | |||||
14 | 4.63 | 172.92 | 6 | |||||
15 | 6.60 | 90.98 | 8 | |||||
16 | 5.40 | 141.30 | 3 | |||||
17 | 6.00 | 36.90 | 5 | |||||
18 | 5.40 | 56.88 | 4 | |||||
19 | 3.36 | 71.82 | 1 | |||||
20 | 4.68 | 69.48 | 3 | |||||
21 | 4.32 | 54.36 | 2 | |||||
22 | 5.52 | 87.66 | 5 | |||||
23 | 4.56 | 38.16 | 3 | |||||
24 | 5.40 | 43.74 | 7 | |||||
25 | 4.78 | 60.72 | 4 | |||||
Food and income are reported in thousands of dollars per year, and the variable size refers to the number of people in the household.
Click here for the Excel Data File
a-1. Develop a correlation matrix. (Round your answers to 3 decimal places. Negative amounts should be indicated by a minus sign.)
a-2. Do you see any problem with multicollinearity?
b-1. Determine the regression equation. (Round your answer to 3 decimal places.)
b-2. How much does an additional family member add to the amount spent on food? (Round your answer to the nearest dollar amount.)
c-1. What is the value of R2? (Round your answer to 3 decimal places.)
c-2. Complete the ANOVA (Leave no cells blank - be certain to enter "0" wherever required. Round SS, MS to 4 decimal places and F to 2 decimal places.)
c-3. State the decision rule for 0.05 significance level. H0: = β1 = β2 = 0; H1: Not all βi's = 0. (Round your answer to 2 decimal places.)
c-4. Can we reject H0: = β1 = β2 = 0?
d-1. Complete the table given below. (Leave no cells blank - be certain to enter "0" wherever required. Round Coefficient, SE Coefficient, P to 4 decimal places and T to 2 decimal places.)
d-2. Would you consider deleting either of the independent variables?
From the graph the residuals appear normally distributed.
True
False
There is a homoscedasticity problem.
There is no homoscedasticity problem.
Solution
a-1
Food | Income | Size | |
Food | 1.000 | ||
Income | 0.107 | 1.000 | |
Size | 0.867 | 0.132 | 1.000 |
a-2. Do you see any problem with multicollinearity?
No Since the correlation between Income and size < 0.5.
b-1. Determine the regression equation.
we will solve it by using excel and the steps are
Enter the Data into excel
Click on Data tab
Click on Data Analysis
Select Regression
Select input Y Range as Range of dependent variable.
Select Input X Range as Range of independent variable
click on labels if your selecting data with labels
click on ok.
So this is the output of Regression in Excel.
SUMMARY OUTPUT | ||||||||
Regression Statistics | ||||||||
Multiple R | 0.8668 | |||||||
R Square | 0.7514 | |||||||
Adjusted R Square | 0.7288 | |||||||
Standard Error | 0.4983 | |||||||
Observations | 25.0000 | |||||||
ANOVA | ||||||||
df | SS | MS | F | Significance F | ||||
Regression | 2.0000 | 16.5115 | 8.2558 | 33.2446 | 0.0000 | |||
Residual | 22.0000 | 5.4633 | 0.2483 | |||||
Total | 24.0000 | 21.9749 | ||||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |
Intercept | 3.3044 | 0.2875 | 11.4923 | 0.0000 | 2.7081 | 3.9007 | 2.7081 | 3.9007 |
Income | -0.0002 | 0.0027 | -0.0733 | 0.9422 | -0.0058 | 0.0054 | -0.0058 | 0.0054 |
Size | 0.4218 | 0.0521 | 8.0922 | 0.0000 | 0.3137 | 0.5298 | 0.3137 | 0.5298 |
Food = 3.304 - 0.000*Income+0.422*Size
b-2. How much does an additional family member add to the amount spent on food?
$0.4218
c-1. What is the value of R2?
R2=0.7514
c-2. Complete the ANOVA
ANOVA | |||||
df | SS | MS | F | Significance F | |
Regression | 2.0000 | 16.5115 | 8.2558 | 33.24 | 0.0000 |
Residual | 22.0000 | 5.4633 | 0.2483 | ||
Total | 24.0000 | 21.9749 |
c-3. State the decision rule for 0.05 significance level. H0: = β1 = β2 = 0; H1: Not all βi's = 0.
If P-value < 0.05 significance level reject the null hypothesis.
c-4. Can we reject H0: = β1 = β2 = 0?
Yes, we reject H0: = β1 = β2 = 0
d-1. Complete the table given below.
Coefficients | Standard Error | t Stat | P-value | |
Intercept | 3.3044 | 0.2875 | 11.49 | 0.0000 |
Income | -0.0002 | 0.0027 | -0.07 | 0.9422 |
Size | 0.4218 | 0.0521 | 8.09 | 0.0000 |
Would you consider deleting either of the independent variables?
Income variable should be deleted.
From the graph the residuals appear normally distributed.
True
There is no homoscedasticity problem